There are a few differences in the two references as the AIM is talking about VFR flights and the .65 doesn't differentiating between VFR and IFR. If you look closely they are not in direct conflict with each other. One says pilots have the responibility to coordinate entry into controlled airspace. The other is saying controller have expectation to do the coordination. The cynic in me says it's typical government double talk. The optimist in me says it worded so that both sides of the mic are vigilant against airspace violations.

In other words, a controller is responsible for coordinating a VFR transition through the clear portion of airspace, but it is incumbent on the pilot to ensure he is cleared into the blue part.

Your example isnt bad but I'm not sure it would apply in practice. If a VFR aircraft is receiving flight following from a radar facility with class C and is going to enter the class c regulatory area, one of two possible scenarios exist as I see it. Either the aircraft will enter the class c regulatory area within the controller's area of jurisdiction and by receiving flight following he has satisfied his requirements for entry into class c. Or he will enter the regulatory area outside the controllers area of jurisdiction and the controller is required to coordinate that. Now in the original post it sounds like the pilot was talking to facililty A while the class c was inside facility B's airspace. Regardless of the type airspace involved I cannot imagine a controller allowing an aircraft receiving services to enter another controllers area of jurisdiction without coordination.

You have a point but let's change it up a bit and imagine the same situation with class B airspace involved. Let's say a center controller is providing VFR flight following and radar advisories to an aircraft which will traverse not only Approach control airspace of jurisdiction, but also Class B airspace. Let's assume the center controller has performed all of the necessary coordination with the approach control regarding the aircraft to transition the airspace including the class B portion. Since a clearance is required for the aircraft to enter class B airspace does that mean that the pilot may not enter the airspace without a clearance by the center controller? I believe the answer is yes, the pilot may not enter class B airspace unless he hears the phrase, "N123, cleared into the XXX class bravo airspace..."

I don't have any experience with class b but I thing you are right. Based on what I'm reading if I were that center controller I would feel obligated to issue that clearance (once the coordination has been accomplished)

I'm not sure that the 7110.65 and AIM are really contradictory. The controller is required to coordinate the transition through a class B, C, or D surface area only if the controller issues a clearance that will require the aircraft to enter that airspace.

Ref 7110.65: 2-1-16. SURFACE AREAS

a. Coordinate with the appropriate nonapproach control tower on an individual aircraft basis before issuing a clearance which would require flight within a surface area for which the tower has responsibility unless otherwise specified in a letter of agreement.

Let me clarify "airspace of jurisdiction" and how that relates to you as a pilot receiving flight following. We all know class B airspace requires a specific clearance to enter so I'm going to ignore that. With class C and D, the surface areas are the airport towers' airspace of jurisdiction. The outer "shelf" of class C airspace and class C outer area would fall into the local approach control airspace of jurisdiction. If you are receiving flight following from a radar facility, the radar controllers will coordinate your transition through the different sectors within approach control/center airspace using normal radar handoff procedures. No action required from the pilot. If the controller specifically clears the pilot to transition the surface area around a towered airport, the controller will either obtain authorization for the pilot to transition through the tower's airspace while remaining on approach frequency, or authorize the pilot to change to the tower frequency. However, if the pilot is maintaining own navigation while receiving flight following, the pilot is responsible for obtaining authorization to transition any surface area and is not implicitly cleared through that airspace just by virtue of receiving flight following. Sorry for the long response. I hope this helps clarify things for you all.

I'm not sure that the 7110.65 and AIM are really contradictory. The controller is required to coordinate the transition through a class B, C, or D surface area only if the controller issues a clearance that will require the aircraft to enter that airspace.

Ref 7110.65: 2-1-16. SURFACE AREAS

a. Coordinate with the appropriate nonapproach control tower on an individual aircraft basis before issuing a clearance which would require flight within a surface area for which the tower has responsibility unless otherwise specified in a letter of agreement.

b. Coordinate with the appropriate control tower for transit authorization when you are providing radar traffic advisory service to an aircraft that will enter another facility's airspace.

NOTE-The pilot is not expected to obtain his/her own authorization through each area when in contact with a radar facility.

Well as I said "semantically" (if that's a word) the AIM and .65 do not contradict however in when you try to apply the two is where the problem comes. I'll come back to that.

In your own post you quote paragraph "b" which is the most applicable as I see it because it's the one that the note applies to.

"b. Coordinate with the appropriate control tower for transit authorization when you are providing radar traffic advisory service to an aircraft that will enter another facility's airspace."

It doesn't say anything about issuing a clearance. Then comes the note to making more clear about what the controller is expected to do.

"NOTE-The pilot is not expected to obtain his/her own authorization through each area when in contact with a radar facility."

Back to the application of the AIM and .65's instructions. If the ATC handbook is telling controllers that pilots aren't expected to do the coordination, then it would make sense that the controllers are the one who are expected to (they're the other party involved in this exchange). And it's hard to imagine that a controller would be expected to do something without being held somehow responsible for doing it.

On the other side of the the radio is the pilot. If an aircraft is receiving flight following from a radar facility and it's flight path it taking it though the D the AIM states it's their responsibility to meet the requirements of entry into controlled airspace. However that would require them to possibly leave the radar facility's frequency, which you can't do without permission, or be actively monitoring two different ATC facilities at the same time which could be very taxing. Then what if the tower gives the aircraft a restriction or instruction which is on conflict an instruction to the radar facility gives or vice versa. That could lead to one big mess. All around it puts the pilot between a rock and hard place

I'll just add one more thing. I don't think this part of your statement is wrong:

the pilot is responsible for obtaining authorization to transition any surface area and is not implicitly cleared through that airspace just by virtue of receiving flight following.

The AIM is pretty clear on that and while I didn't look up the FAR it referenced I'm guessing it says the same. But I don't think there is any differentiation between in the controller's responsibility in when a clearance as been issued and simple flight following is being given.